The mathematical model is developed to investigate the relationship between the Hamiltonian of the Proton Exchange Membrane Fuel Cell (PEMFC) and the output of the PEMFC. At constant response time, the Hamiltonian is directly proportional to current density and cell voltage, according to the established model (Nano-second). At activation overpotential 90mV, the Hamiltonian of PEMFC with current density increases without a hump, however at activation overpotential 1mV, a bump is detected at 0.47mAcm-2. The hump is also seen with cell voltage and power density at the same current density. The current density exponential term is responsible for the hump. PEMFC's Hamiltonian at low temperature is higher than PEMFC's Hamiltonian at high temperature. By examining the Hamiltonian with temperature and its relationship to a PEMFC output, the efficiency of the PEMFC can be increased by minimizing heat. Because PEMFC generates electricity and heat as a result of the chemical process. The collision of free electrons and protons around the electrodes generates heat energy. Because the Hamiltonian is inversely proportional to distance and directly proportional to current density and power density, the energy carried by free electrons and proton decreases as the distance between them decreases, resulting in the generation of heat energy during collisionless collisions. The free electrons and free protons circulate faster to form PEMFC outputs due to the short distance gap.

This paper deals with S-matrix, born first approximation, amplitude, and differential cross-section (DCS), using Volkov function and Taylor series expansion in laser field, scattering. Equation (30) copes-with DCS and Equation (36) deals with S-matrix, with different parameters, moreover, both equations contain real and imaginary parts. The DCS increases with increasing angle and polarizabilities, constant with dipole distance for both emission and absorption of single-photon. The DCS for both emission and absorption is responded to low incidence energy (30 eV -60 eV) and photon energy (15 eV) while at high energy only emission and absorption are responded for DCS. The DCS between absorption and emission of a photon with angle variation, dipole distance, and atomic polarizabilities was found 1.098 a.u. 2 and at high incidence, energies were found 0.1 a.u. 2 .